# Fourier series of a continuous function converging to non-value of function?

I know of examples of periodic continuous functions whose Fourier series diverge on sets of measure zero. Is it possible for a Fourier series of a periodic continuous function to converge to something other than the function value, at some point? If so, can someone provide an example or reference?

1. If a series converges to a limit $L$, its Cesaro averages converge to the same limit $L$.
2. The Cesaro averages of a Fourier series are given by convolution with the Fejer kernel, and it is known that convolving an element of $C({\bf T})$ with the Fejer kernel always gives a sequence of trigonometric polynomials converging uniformly to the given function.
So I think the answer to your question is negative: if $f$ is continuous, and if the partial sums of $\sum_{n\in{\bf Z}} \hat{f}(n)e^{2\pi int}$ converge to some $L$, then $L=f(t)$.